The discrete Pompeiu problem on the plane · 2018. 4. 6. · Dimitrie Pompeiu (1873-1954) was a...

The discrete Pompeiu problem on the plane

Gergely Kiss

Joint work with M. Laczkovich and Cs. Vincze

University of Luxembourg

DCG-DISOPT seminarLausanne, November 27, 2017

Dimitrie Pompeiu (1873-1954) was a Romanian mathematician. Asone of the students of Henri Poincare he obtained a Ph.D. degreein mathematics in 1905 at the Universite de Paris (Sorbonne).

His contributions were mainly connected to the fields ofmathematical analysis, especially the theory of complex functions.In one of his article1 he posed a question of integral geometry.

It iswidely known as the Pompeiu problem.

1Sur certains systemes d’equations lineaires et sur une propriete integraledes fonctions de plusieurs variables (Comptes Rendus de l’Academie desSciences. Serie I. Mathematique 188 (1929) pp. 1138-1139)

Dimitrie Pompeiu (1873-1954) was a Romanian mathematician. Asone of the students of Henri Poincare he obtained a Ph.D. degreein mathematics in 1905 at the Universite de Paris (Sorbonne).

His contributions were mainly connected to the fields ofmathematical analysis, especially the theory of complex functions.In one of his article1 he posed a question of integral geometry. It iswidely known as the Pompeiu problem.

1Sur certains systemes d’equations lineaires et sur une propriete integraledes fonctions de plusieurs variables (Comptes Rendus de l’Academie desSciences. Serie I. Mathematique 188 (1929) pp. 1138-1139)

The classical Pompeiu problem

QuestionLet f be a continuous function defined on the plane, and let K bea closed set of positive Lebesgue measure. Suppose that∫

σ(K)f (x , y)dλxdλy = 0 (1)

for every rigid motion σ of the plane, where λ denotes theLebesque measure. Is it true that f ≡ 0?

We say that the set K has the Pompeiu’s property if the answerof this question is affirmative (YES).

The classical Pompeiu problem

QuestionLet f be a continuous function defined on the plane, and let K bea closed set of positive Lebesgue measure. Suppose that∫

σ(K)f (x , y)dλxdλy = 0 (1)

for every rigid motion σ of the plane, where λ denotes theLebesque measure. Is it true that f ≡ 0?

We say that the set K has the Pompeiu’s property if the answerof this question is affirmative (YES).

The continuous case

1. K = square:I Pompeiu showed that the square has the Pompeiu property.

I Brown, Schreiber, Taylor2 generalized this:Every convex set K has the Pompieu property if it has at leastone ”corner”.

2. K = disk:I Chakalov3 showed infinitely many linearly independent

solutions of the form sin(ax + by) for appropriately chosenconstants a, b.

I Williams4 proved that if K is simply-connected and has asufficiently smooth boundary ∂K then there is a functionf 6≡ 0.

2L. Brown, B. M. Schreiber and B. A. Taylor, Spectral synthesis and thePompeiu problem, Ann. Inst. Fourier 23 (1973), 125-154.

3Chalakov, Sur un probleme de D. Pompeiu, Annaire Univ. Sofia Fac. Phys.Math., Livre 1, 40 (1944), 1-44.

4S. Williams, A partial solutions of Pompeiu problem, Math. Ann. 223(1976), 183-190.

The continuous case

1. K = square:I Pompeiu showed that the square has the Pompeiu property.I Brown, Schreiber, Taylor2 generalized this:

Every convex set K has the Pompieu property if it has at leastone ”corner”.

2. K = disk:I Chakalov3 showed infinitely many linearly independent

solutions of the form sin(ax + by) for appropriately chosenconstants a, b.

I Williams4 proved that if K is simply-connected and has asufficiently smooth boundary ∂K then there is a functionf 6≡ 0.

2L. Brown, B. M. Schreiber and B. A. Taylor, Spectral synthesis and thePompeiu problem, Ann. Inst. Fourier 23 (1973), 125-154.

3Chalakov, Sur un probleme de D. Pompeiu, Annaire Univ. Sofia Fac. Phys.Math., Livre 1, 40 (1944), 1-44.

4S. Williams, A partial solutions of Pompeiu problem, Math. Ann. 223(1976), 183-190.

The continuous case

1. K = square:I Pompeiu showed that the square has the Pompeiu property.I Brown, Schreiber, Taylor2 generalized this:

Every convex set K has the Pompieu property if it has at leastone ”corner”.

2. K = disk:I Chakalov3 showed infinitely many linearly independent

solutions of the form sin(ax + by) for appropriately chosenconstants a, b.

I Williams4 proved that if K is simply-connected and has asufficiently smooth boundary ∂K then there is a functionf 6≡ 0.

2L. Brown, B. M. Schreiber and B. A. Taylor, Spectral synthesis and thePompeiu problem, Ann. Inst. Fourier 23 (1973), 125-154.

3Chalakov, Sur un probleme de D. Pompeiu, Annaire Univ. Sofia Fac. Phys.Math., Livre 1, 40 (1944), 1-44.

4S. Williams, A partial solutions of Pompeiu problem, Math. Ann. 223(1976), 183-190.

The continuous case

1. K = square:I Pompeiu showed that the square has the Pompeiu property.I Brown, Schreiber, Taylor2 generalized this:

Every convex set K has the Pompieu property if it has at leastone ”corner”.

2. K = disk:I Chakalov3 showed infinitely many linearly independent

solutions of the form sin(ax + by) for appropriately chosenconstants a, b.

I Williams4 proved that if K is simply-connected and has asufficiently smooth boundary ∂K then there is a functionf 6≡ 0.

2L. Brown, B. M. Schreiber and B. A. Taylor, Spectral synthesis and thePompeiu problem, Ann. Inst. Fourier 23 (1973), 125-154.

3Chalakov, Sur un probleme de D. Pompeiu, Annaire Univ. Sofia Fac. Phys.Math., Livre 1, 40 (1944), 1-44.

4S. Williams, A partial solutions of Pompeiu problem, Math. Ann. 223(1976), 183-190.

Another type of problem - The discrete case

I 70th Putman Mathematical Competition problem (2009):Let f be a real-valued function on the plane such that forevery square ABCD in the plane,f (A) + f (B) + f (C ) + f (D) = 0. Does it follow that f (P) = 0for all points P in the plane?

I Similar question is whether f ≡ 0 wheneverf (A) + f (B) + f (C ) + f (D) = 0 holds for every unit squareABCD. Katz, Krebs, Shaheen5 shown a nice elementary proof.

I Groote, Duerinckx6 investigated the following version: Given anonempty finite set H ⊂ R2 and a function f : R2 → Rd suchthat the arithmetic mean of f at the elements of any similarcopy of H is constant. Does it follow that f is constant onR2?

5R. Katz, M. Krebs, A. Shaheen, Zero sums on unit square vertex sets andplane colorings, Amer. Math. Monthly 121 (2014), no. 7, 610–618.

6C. de Groote, M. Duerinckx, Functions with constant mean on similarcountable subsets of R2, Amer. Math. Monthly 119 (2012), 603–605.

Another type of problem - The discrete case

I 70th Putman Mathematical Competition problem (2009):Let f be a real-valued function on the plane such that forevery square ABCD in the plane,f (A) + f (B) + f (C ) + f (D) = 0. Does it follow that f (P) = 0for all points P in the plane?

I Similar question is whether f ≡ 0 wheneverf (A) + f (B) + f (C ) + f (D) = 0 holds for every unit squareABCD. Katz, Krebs, Shaheen5 shown a nice elementary proof.

I Groote, Duerinckx6 investigated the following version: Given anonempty finite set H ⊂ R2 and a function f : R2 → Rd suchthat the arithmetic mean of f at the elements of any similarcopy of H is constant. Does it follow that f is constant onR2?

5R. Katz, M. Krebs, A. Shaheen, Zero sums on unit square vertex sets andplane colorings, Amer. Math. Monthly 121 (2014), no. 7, 610–618.

6C. de Groote, M. Duerinckx, Functions with constant mean on similarcountable subsets of R2, Amer. Math. Monthly 119 (2012), 603–605.

Another type of problem - The discrete case

I 70th Putman Mathematical Competition problem (2009):Let f be a real-valued function on the plane such that forevery square ABCD in the plane,f (A) + f (B) + f (C ) + f (D) = 0. Does it follow that f (P) = 0for all points P in the plane?

I Similar question is whether f ≡ 0 wheneverf (A) + f (B) + f (C ) + f (D) = 0 holds for every unit squareABCD. Katz, Krebs, Shaheen5 shown a nice elementary proof.

I Groote, Duerinckx6 investigated the following version: Given anonempty finite set H ⊂ R2 and a function f : R2 → Rd suchthat the arithmetic mean of f at the elements of any similarcopy of H is constant. Does it follow that f is constant onR2?

5R. Katz, M. Krebs, A. Shaheen, Zero sums on unit square vertex sets andplane colorings, Amer. Math. Monthly 121 (2014), no. 7, 610–618.

6C. de Groote, M. Duerinckx, Functions with constant mean on similarcountable subsets of R2, Amer. Math. Monthly 119 (2012), 603–605.

Another type of problem - The discrete case

I 70th Putman Mathematical Competition problem (2009):Let f be a real-valued function on the plane such that forevery square ABCD in the plane,f (A) + f (B) + f (C ) + f (D) = 0. Does it follow that f (P) = 0for all points P in the plane?

I Similar question is whether f ≡ 0 wheneverf (A) + f (B) + f (C ) + f (D) = 0 holds for every unit squareABCD. Katz, Krebs, Shaheen5 shown a nice elementary proof.

I Groote, Duerinckx6 investigated the following version: Given anonempty finite set H ⊂ R2 and a function f : R2 → Rd suchthat the arithmetic mean of f at the elements of any similarcopy of H is constant. Does it follow that f is constant onR2?

5R. Katz, M. Krebs, A. Shaheen, Zero sums on unit square vertex sets andplane colorings, Amer. Math. Monthly 121 (2014), no. 7, 610–618.

6C. de Groote, M. Duerinckx, Functions with constant mean on similarcountable subsets of R2, Amer. Math. Monthly 119 (2012), 603–605.

More abstractly

Let X be a nonempty set and G be the family of bijectionsmapping X to itself.

DefinitionThe finite set H = {d1, . . . , dn} ⊂ X has the discrete Pompeiuproperty w.r.t. G if for every function f : X → C the equation

n∑i=1

f (σ(di )) = 0 (2)

holds for all σ ∈ G implies that f ≡ 0.

In the definition we replace the Lebesque measure by the countingmeasure.

More abstractly

Let X be a nonempty set and G be the family of bijectionsmapping X to itself.

DefinitionThe finite set H = {d1, . . . , dn} ⊂ X has the discrete Pompeiuproperty w.r.t. G if for every function f : X → C the equation

n∑i=1

f (σ(di )) = 0 (2)

holds for all σ ∈ G implies that f ≡ 0.

In the definition we replace the Lebesque measure by the countingmeasure.

Weighted version

DefinitionThe finite set H = {d1, . . . , dn} ⊂ X has the weighted discretePompeiu property w.r.t. G if the following condition satisfied:whenever a1, . . . , an are complex numbers with

∑ni=1 ai 6= 0 and

f : X → C, the equation

n∑i=1

ai f (σ(di )) = 0 (3)

holds for all σ ∈ G, then f ≡ 0.

We focus on the similarity group (S), the translation group (T )and the isometry group (I) of C.

Weighted version

DefinitionThe finite set H = {d1, . . . , dn} ⊂ X has the weighted discretePompeiu property w.r.t. G if the following condition satisfied:whenever a1, . . . , an are complex numbers with

∑ni=1 ai 6= 0 and

f : X → C, the equation

n∑i=1

ai f (σ(di )) = 0 (3)

holds for all σ ∈ G, then f ≡ 0.

We focus on the similarity group (S), the translation group (T )and the isometry group (I) of C.

Similarity case

TheoremEvery finite set H has the weighted discrete Pompeiu propertyw.r.t. S.

Let d1, d2, . . . ,dn ∈ C be the points of H. Then the equation (3)can be written in the form

a1f (x + d1y) + a2f (x + d2y) + . . .+ anf (x + dny) = 0 (4)

for every x , y ∈ C, y 6= 0 with constant a1, . . . ,an ∈ C with∑ni=1 ai 6= 0.

Theorem (K.- Varga7, ’14)

Assume that (4) holds for every x , y ∈ C. Then∃f 6≡ 0 solution of (4) ⇐⇒

⇐⇒ ∃ automorphism φ of C which is a solution of (4) ⇐⇒⇐⇒ φ satisfies

∑i ai = 0 and

∑i aiφ(bi ) = 0.

7G. Kiss and A. Varga, Existence of nontrivial solutions of linear functionalequation, Aequationes Math. (2014), 88, no. 1, 151-162.

Similarity case

TheoremEvery finite set H has the weighted discrete Pompeiu propertyw.r.t. S.

Let d1, d2, . . . ,dn ∈ C be the points of H. Then the equation (3)can be written in the form

a1f (x + d1y) + a2f (x + d2y) + . . .+ anf (x + dny) = 0 (4)

for every x , y ∈ C, y 6= 0 with constant a1, . . . ,an ∈ C with∑ni=1 ai 6= 0.

Theorem (K.- Varga7, ’14)

Assume that (4) holds for every x , y ∈ C. Then∃f 6≡ 0 solution of (4) ⇐⇒

⇐⇒ ∃ automorphism φ of C which is a solution of (4) ⇐⇒⇐⇒ φ satisfies

∑i ai = 0 and

∑i aiφ(bi ) = 0.

7G. Kiss and A. Varga, Existence of nontrivial solutions of linear functionalequation, Aequationes Math. (2014), 88, no. 1, 151-162.

Similarity case

TheoremEvery finite set H has the weighted discrete Pompeiu propertyw.r.t. S.

Let d1, d2, . . . ,dn ∈ C be the points of H. Then the equation (3)can be written in the form

a1f (x + d1y) + a2f (x + d2y) + . . .+ anf (x + dny) = 0 (4)

for every x , y ∈ C, y 6= 0 with constant a1, . . . ,an ∈ C with∑ni=1 ai 6= 0.

Theorem (K.- Varga7, ’14)

Assume that (4) holds for every x , y ∈ C. Then∃f 6≡ 0 solution of (4) ⇐⇒

⇐⇒ ∃ automorphism φ of C which is a solution of (4) ⇐⇒⇐⇒ φ satisfies

∑i ai = 0 and

∑i aiφ(bi ) = 0.

7G. Kiss and A. Varga, Existence of nontrivial solutions of linear functionalequation, Aequationes Math. (2014), 88, no. 1, 151-162.

Slight generalizationWe say that a family I of subsets of C is proper and translationinvariant ideal, if

I A,B ∈ I implies that A ∪ B ∈ I

I A ∈ I and B ⊆ A, then B ∈ I

I C 6∈ I

I A ∈ I , then A + c = {x + c : x ∈ A} ∈ I for every c ∈ I .

Example

I can be the finite subsets of C, all sets of measure 0 or all firstcategory sets of C.

TheoremLet I be a proper and translation invariant ideal. Assume that (4)holds for every x ∈ C, y ∈ C \ A, where A ∈ I . Then

∃f 6≡ 0 solution of (4) ⇐⇒⇐⇒ ∃ automorphism φ of C which is a solution of (4) ⇐⇒

⇐⇒ φ satisfies∑

i ai = 0 and∑

i aiφ(bi ) = 0.

Slight generalizationWe say that a family I of subsets of C is proper and translationinvariant ideal, if

I A,B ∈ I implies that A ∪ B ∈ I

I A ∈ I and B ⊆ A, then B ∈ I

I C 6∈ I

I A ∈ I , then A + c = {x + c : x ∈ A} ∈ I for every c ∈ I .

Example

I can be the finite subsets of C, all sets of measure 0 or all firstcategory sets of C.

TheoremLet I be a proper and translation invariant ideal. Assume that (4)holds for every x ∈ C, y ∈ C \ A, where A ∈ I . Then

∃f 6≡ 0 solution of (4) ⇐⇒⇐⇒ ∃ automorphism φ of C which is a solution of (4) ⇐⇒

⇐⇒ φ satisfies∑

i ai = 0 and∑

i aiφ(bi ) = 0.

Slight generalizationWe say that a family I of subsets of C is proper and translationinvariant ideal, if

I A,B ∈ I implies that A ∪ B ∈ I

I A ∈ I and B ⊆ A, then B ∈ I

I C 6∈ I

I A ∈ I , then A + c = {x + c : x ∈ A} ∈ I for every c ∈ I .

Example

I can be the finite subsets of C, all sets of measure 0 or all firstcategory sets of C.

TheoremLet I be a proper and translation invariant ideal. Assume that (4)holds for every x ∈ C, y ∈ C \ A, where A ∈ I . Then

∃f 6≡ 0 solution of (4) ⇐⇒⇐⇒ ∃ automorphism φ of C which is a solution of (4) ⇐⇒

⇐⇒ φ satisfies∑

i ai = 0 and∑

i aiφ(bi ) = 0.

Slight generalizationWe say that a family I of subsets of C is proper and translationinvariant ideal, if

I A,B ∈ I implies that A ∪ B ∈ I

I A ∈ I and B ⊆ A, then B ∈ I

I C 6∈ I

I A ∈ I , then A + c = {x + c : x ∈ A} ∈ I for every c ∈ I .

Example

I can be the finite subsets of C, all sets of measure 0 or all firstcategory sets of C.

TheoremLet I be a proper and translation invariant ideal. Assume that (4)holds for every x ∈ C, y ∈ C \ A, where A ∈ I . Then

∃f 6≡ 0 solution of (4) ⇐⇒⇐⇒ ∃ automorphism φ of C which is a solution of (4) ⇐⇒

⇐⇒ φ satisfies∑

i ai = 0 and∑

i aiφ(bi ) = 0.

Translations

TheoremLet G be a torsion free Abelian group. No finite setH ⊂ G , |H| ≥ 2 has the discrete Pompeiu property w.r.t. G .

Let GH be the subgroup of G generated by H. Then GH is afinitely generated torsion free Abelian group, and thus

GH∼= Zn for some finite n.

Theorem (Zeilberger8)

For every finite set H ⊂ Zn there is a nonzero function f : Zn → Csuch that

∑ni=1 f (σ(di )) = 0 for any translate σ of Zn.

Easily, we can find such a function on every coset of G/GH .

8D. Zeilberger, Pompeiu’s problem in discrete space, Proc. Nat. Acad. Sci.USA 75 (1978), no. 8, 3555-3556.

Translations

TheoremLet G be a torsion free Abelian group. No finite setH ⊂ G , |H| ≥ 2 has the discrete Pompeiu property w.r.t. G .

Let GH be the subgroup of G generated by H. Then GH is afinitely generated torsion free Abelian group, and thus

GH∼= Zn for some finite n.

Theorem (Zeilberger8)

For every finite set H ⊂ Zn there is a nonzero function f : Zn → Csuch that

∑ni=1 f (σ(di )) = 0 for any translate σ of Zn.

Easily, we can find such a function on every coset of G/GH .

8D. Zeilberger, Pompeiu’s problem in discrete space, Proc. Nat. Acad. Sci.USA 75 (1978), no. 8, 3555-3556.

Translations

TheoremLet G be a torsion free Abelian group. No finite setH ⊂ G , |H| ≥ 2 has the discrete Pompeiu property w.r.t. G .

Let GH be the subgroup of G generated by H. Then GH is afinitely generated torsion free Abelian group, and thus

GH∼= Zn for some finite n.

Theorem (Zeilberger8)

For every finite set H ⊂ Zn there is a nonzero function f : Zn → Csuch that

∑ni=1 f (σ(di )) = 0 for any translate σ of Zn.

Easily, we can find such a function on every coset of G/GH .

8D. Zeilberger, Pompeiu’s problem in discrete space, Proc. Nat. Acad. Sci.USA 75 (1978), no. 8, 3555-3556.

Translations

TheoremLet G be a torsion free Abelian group. No finite setH ⊂ G , |H| ≥ 2 has the discrete Pompeiu property w.r.t. G .

Let GH be the subgroup of G generated by H. Then GH is afinitely generated torsion free Abelian group, and thus

GH∼= Zn for some finite n.

Theorem (Zeilberger8)

For every finite set H ⊂ Zn there is a nonzero function f : Zn → Csuch that

∑ni=1 f (σ(di )) = 0 for any translate σ of Zn.

Easily, we can find such a function on every coset of G/GH .

8D. Zeilberger, Pompeiu’s problem in discrete space, Proc. Nat. Acad. Sci.USA 75 (1978), no. 8, 3555-3556.

Isometries

Proposition

Let E be a finite set in Rn. If there exists an isometry σ such that|E ∩ σ(E )| = |E | − 1, then E has the discrete Pompeiu propertyw.r.t. isometries.

Corollary

For the vertex set of any k - dimensional simplex in Rn (n ≥ 2) hasthe discrete Pompeiu property w.r.t. I.

Proposition

Every 2- and 3-element set has the discrete weighted Pompeiuproperty w.r.t. I.

Isometries

Proposition

Let E be a finite set in Rn. If there exists an isometry σ such that|E ∩ σ(E )| = |E | − 1, then E has the discrete Pompeiu propertyw.r.t. isometries.

Corollary

For the vertex set of any k - dimensional simplex in Rn (n ≥ 2) hasthe discrete Pompeiu property w.r.t. I.

Proposition

Every 2- and 3-element set has the discrete weighted Pompeiuproperty w.r.t. I.

Isometries

Proposition

Let E be a finite set in Rn. If there exists an isometry σ such that|E ∩ σ(E )| = |E | − 1, then E has the discrete Pompeiu propertyw.r.t. isometries.

Corollary

For the vertex set of any k - dimensional simplex in Rn (n ≥ 2) hasthe discrete Pompeiu property w.r.t. I.

Proposition

Every 2- and 3-element set has the discrete weighted Pompeiuproperty w.r.t. I.

Main results9

TheoremLet D be the vertex set of any parallelogram. Then D has thediscrete Pompeiu property w.r.t. I.

TheoremLet D ⊂ R2 be the set of four points with rational coordinates.Then D has the discrete weighted Pompeiu property w.r.t. I.

TheoremLet D be an n-tuple of collinear points in the plane with pairwisecommensurable distances. Then D has the discrete weightedPompeiu property w.r.t. I.

9G. Kiss, M. Laczkovich and Cs. Vincze, The discrete Pompeiu problem onthe plane, accepted at Monatshefte fur Mathematik,https://arxiv.org/pdf/1612.00284v1.pdf

Main results9

TheoremLet D be the vertex set of any parallelogram. Then D has thediscrete Pompeiu property w.r.t. I.

TheoremLet D ⊂ R2 be the set of four points with rational coordinates.Then D has the discrete weighted Pompeiu property w.r.t. I.

TheoremLet D be an n-tuple of collinear points in the plane with pairwisecommensurable distances. Then D has the discrete weightedPompeiu property w.r.t. I.

9G. Kiss, M. Laczkovich and Cs. Vincze, The discrete Pompeiu problem onthe plane, accepted at Monatshefte fur Mathematik,https://arxiv.org/pdf/1612.00284v1.pdf

Main results9

TheoremLet D be the vertex set of any parallelogram. Then D has thediscrete Pompeiu property w.r.t. I.

TheoremLet D ⊂ R2 be the set of four points with rational coordinates.Then D has the discrete weighted Pompeiu property w.r.t. I.

TheoremLet D be an n-tuple of collinear points in the plane with pairwisecommensurable distances. Then D has the discrete weightedPompeiu property w.r.t. I.

9G. Kiss, M. Laczkovich and Cs. Vincze, The discrete Pompeiu problem onthe plane, accepted at Monatshefte fur Mathematik,https://arxiv.org/pdf/1612.00284v1.pdf

Open Questions

A general open question is the following:

QuestionAre there any finite set in the plane which has not the discretePompeiu property w.r.t. I?

QuestionDo the following sets have the discrete Pompeiu property

1. Symmetric trapezoid,

2. 4 points in a line,

3. Pentagon, regular n-gon?

Open Questions

A general open question is the following:

QuestionAre there any finite set in the plane which has not the discretePompeiu property w.r.t. I?

QuestionDo the following sets have the discrete Pompeiu property

1. Symmetric trapezoid,

2. 4 points in a line,

3. Pentagon, regular n-gon?

Spectral analysis on discrete Abelian groups

In the continuous case the proofs based on harmonic analysis.

Analogously, let G be an Abelian group and we introduce:

1. CG = {f : G → C} the linear space of all complex valuedfunctions defined on G equipped with the product topology.

2. Exponential function g : G → C:g 6= 0 and g(x + y) = g(x) · g(y) for every x , y ∈ G .

3. Variety V on G :

3.1 translation invariant3.2 closed3.3 linear

subspace of CG .

4. Spectral analysis holds in G : every V 6= {0} on G contains anexponential function.

Spectral analysis on discrete Abelian groups

In the continuous case the proofs based on harmonic analysis.Analogously, let G be an Abelian group and we introduce:

1. CG = {f : G → C} the linear space of all complex valuedfunctions defined on G equipped with the product topology.

2. Exponential function g : G → C:g 6= 0 and g(x + y) = g(x) · g(y) for every x , y ∈ G .

3. Variety V on G :

3.1 translation invariant3.2 closed3.3 linear

subspace of CG .

4. Spectral analysis holds in G : every V 6= {0} on G contains anexponential function.

Spectral analysis on discrete Abelian groups

In the continuous case the proofs based on harmonic analysis.Analogously, let G be an Abelian group and we introduce:

1. CG = {f : G → C} the linear space of all complex valuedfunctions defined on G equipped with the product topology.

2. Exponential function g : G → C:g 6= 0 and g(x + y) = g(x) · g(y) for every x , y ∈ G .

3. Variety V on G :

3.1 translation invariant3.2 closed3.3 linear

subspace of CG .

4. Spectral analysis holds in G : every V 6= {0} on G contains anexponential function.

Spectral analysis on discrete Abelian groups

In the continuous case the proofs based on harmonic analysis.Analogously, let G be an Abelian group and we introduce:

1. CG = {f : G → C} the linear space of all complex valuedfunctions defined on G equipped with the product topology.

2. Exponential function g : G → C:g 6= 0 and g(x + y) = g(x) · g(y) for every x , y ∈ G .

3. Variety V on G :

3.1 translation invariant3.2 closed3.3 linear

subspace of CG .

4. Spectral analysis holds in G : every V 6= {0} on G contains anexponential function.

Spectral analysis on discrete Abelian groups

In the continuous case the proofs based on harmonic analysis.Analogously, let G be an Abelian group and we introduce:

1. CG = {f : G → C} the linear space of all complex valuedfunctions defined on G equipped with the product topology.

2. Exponential function g : G → C:g 6= 0 and g(x + y) = g(x) · g(y) for every x , y ∈ G .

3. Variety V on G :

3.1 translation invariant3.2 closed3.3 linear

subspace of CG .

4. Spectral analysis holds in G : every V 6= {0} on G contains anexponential function.

Spectral analysis on discrete Abelian groups

In the continuous case the proofs based on harmonic analysis.Analogously, let G be an Abelian group and we introduce:

1. CG = {f : G → C} the linear space of all complex valuedfunctions defined on G equipped with the product topology.

2. Exponential function g : G → C:g 6= 0 and g(x + y) = g(x) · g(y) for every x , y ∈ G .

3. Variety V on G :

3.1 translation invariant3.2 closed3.3 linear

subspace of CG .

4. Spectral analysis holds in G : every V 6= {0} on G contains anexponential function.

The torsion free rank r0(G ) of G is the cardinality of a maximalindependent system of elements of infinite order.

Theorem (Laczkovich, Szekelyhidi10)

Spectral analysis holds on every Abelian group G iff

r0(G ) < c.

Every function f satisfying

n∑i=1

ai f (x + diy) = 0,

where x , y ∈ C, |y | = 1,∑n

i=1 ai 6= 0 and H = {d1, d2, . . . , dn},constructs a variety V on the additive group of C.

10M. Laczkovich and G. Szekelyhidi, Harmonic analysis on discrete Abeliangroups, Proc. Am. Math. Soc. 133 (2004), no. 6, 1581-1586.

The torsion free rank r0(G ) of G is the cardinality of a maximalindependent system of elements of infinite order.

Theorem (Laczkovich, Szekelyhidi10)

Spectral analysis holds on every Abelian group G iff

r0(G ) < c.

Every function f satisfying

n∑i=1

ai f (x + diy) = 0,

where x , y ∈ C, |y | = 1,∑n

i=1 ai 6= 0 and H = {d1, d2, . . . , dn},constructs a variety V on the additive group of C.

10M. Laczkovich and G. Szekelyhidi, Harmonic analysis on discrete Abeliangroups, Proc. Am. Math. Soc. 133 (2004), no. 6, 1581-1586.

The torsion free rank r0(G ) of G is the cardinality of a maximalindependent system of elements of infinite order.

Theorem (Laczkovich, Szekelyhidi10)

Spectral analysis holds on every Abelian group G iff

r0(G ) < c.

Every function f satisfying

n∑i=1

ai f (x + diy) = 0,

where x , y ∈ C, |y | = 1,∑n

i=1 ai 6= 0 and H = {d1, d2, . . . , dn},constructs a variety V on the additive group of C.

10M. Laczkovich and G. Szekelyhidi, Harmonic analysis on discrete Abeliangroups, Proc. Am. Math. Soc. 133 (2004), no. 6, 1581-1586.

Method of the proof

TheoremLet D be the vertex set of any parallelogram.Then D has the discrete Pompeiu property w.r.t. I.

Proof.

Let G be a finitely generated additive subgroup of C. Thestatement can be written in the following form:

f (x) + f (x + s1y) + f (x + s2y) + f (x + (s1 + s2)y) = 0 (5)

holds for every x , y ∈ G and |y | = 1.Assume that f (0) 6= 0.There exists an exponential function g 6≡ 0 in V which satisfies

g(x + y) = g(x)g(y). (6)

Thus, we get(1 + g(s1y))(1 + g(s2y)) = 0. (7)

i.e g(s1y) = −1 or g(s2y) = −1 (∀y ∈ G , |y | = 1).

Method of the proof

TheoremLet D be the vertex set of any parallelogram.Then D has the discrete Pompeiu property w.r.t. I.

Proof.Let G be a finitely generated additive subgroup of C.

Thestatement can be written in the following form:

f (x) + f (x + s1y) + f (x + s2y) + f (x + (s1 + s2)y) = 0 (5)

holds for every x , y ∈ G and |y | = 1.Assume that f (0) 6= 0.There exists an exponential function g 6≡ 0 in V which satisfies

g(x + y) = g(x)g(y). (6)

Thus, we get(1 + g(s1y))(1 + g(s2y)) = 0. (7)

i.e g(s1y) = −1 or g(s2y) = −1 (∀y ∈ G , |y | = 1).

Method of the proof

TheoremLet D be the vertex set of any parallelogram.Then D has the discrete Pompeiu property w.r.t. I.

Proof.Let G be a finitely generated additive subgroup of C. Thestatement can be written in the following form:

f (x) + f (x + s1y) + f (x + s2y) + f (x + (s1 + s2)y) = 0 (5)

holds for every x , y ∈ G and |y | = 1.

Assume that f (0) 6= 0.There exists an exponential function g 6≡ 0 in V which satisfies

g(x + y) = g(x)g(y). (6)

Thus, we get(1 + g(s1y))(1 + g(s2y)) = 0. (7)

i.e g(s1y) = −1 or g(s2y) = −1 (∀y ∈ G , |y | = 1).

Method of the proof

TheoremLet D be the vertex set of any parallelogram.Then D has the discrete Pompeiu property w.r.t. I.

Proof.Let G be a finitely generated additive subgroup of C. Thestatement can be written in the following form:

f (x) + f (x + s1y) + f (x + s2y) + f (x + (s1 + s2)y) = 0 (5)

holds for every x , y ∈ G and |y | = 1.Assume that f (0) 6= 0.

There exists an exponential function g 6≡ 0 in V which satisfies

g(x + y) = g(x)g(y). (6)

Thus, we get(1 + g(s1y))(1 + g(s2y)) = 0. (7)

i.e g(s1y) = −1 or g(s2y) = −1 (∀y ∈ G , |y | = 1).

Method of the proof

TheoremLet D be the vertex set of any parallelogram.Then D has the discrete Pompeiu property w.r.t. I.

Proof.Let G be a finitely generated additive subgroup of C. Thestatement can be written in the following form:

f (x) + f (x + s1y) + f (x + s2y) + f (x + (s1 + s2)y) = 0 (5)

holds for every x , y ∈ G and |y | = 1.Assume that f (0) 6= 0.There exists an exponential function g 6≡ 0 in V which satisfies

g(x + y) = g(x)g(y). (6)

Thus, we get(1 + g(s1y))(1 + g(s2y)) = 0. (7)

i.e g(s1y) = −1 or g(s2y) = −1 (∀y ∈ G , |y | = 1).

Method of the proof

TheoremLet D be the vertex set of any parallelogram.Then D has the discrete Pompeiu property w.r.t. I.

Proof.Let G be a finitely generated additive subgroup of C. Thestatement can be written in the following form:

f (x) + f (x + s1y) + f (x + s2y) + f (x + (s1 + s2)y) = 0 (5)

holds for every x , y ∈ G and |y | = 1.Assume that f (0) 6= 0.There exists an exponential function g 6≡ 0 in V which satisfies

g(x + y) = g(x)g(y). (6)

Thus, we get(1 + g(s1y))(1 + g(s2y)) = 0. (7)

i.e g(s1y) = −1 or g(s2y) = −1 (∀y ∈ G , |y | = 1).

Method of the proof

TheoremLet D be the vertex set of any parallelogram.Then D has the discrete Pompeiu property w.r.t. I.

Proof.Let G be a finitely generated additive subgroup of C. Thestatement can be written in the following form:

f (x) + f (x + s1y) + f (x + s2y) + f (x + (s1 + s2)y) = 0 (5)

holds for every x , y ∈ G and |y | = 1.Assume that f (0) 6= 0.There exists an exponential function g 6≡ 0 in V which satisfies

g(x + y) = g(x)g(y). (6)

Thus, we get(1 + g(s1y))(1 + g(s2y)) = 0. (7)

i.e g(s1y) = −1 or g(s2y) = −1 (∀y ∈ G , |y | = 1).

a = x , b = x + s1y ,

d = x + s2y , c = x + (s1 + s2)y ,

Then g at the points a, b, c , d are

either g(a),−g(a),−g(d), g(d)

or g(a), g(b),−g(b),−g(a),

respectively.Let A and B be such that C \ {0} = C∗ = A ∪∗ B and A = −B.We define h : C→ {−1, 1} as follows:

h(x) =

{1, if g(x) ∈ A

−1, if g(x) ∈ B.

a = x , b = x + s1y ,

d = x + s2y , c = x + (s1 + s2)y ,

Then g at the points a, b, c , d are

either g(a),−g(a),−g(d), g(d)

or g(a), g(b),−g(b),−g(a),

respectively.

Let A and B be such that C \ {0} = C∗ = A ∪∗ B and A = −B.We define h : C→ {−1, 1} as follows:

h(x) =

{1, if g(x) ∈ A

−1, if g(x) ∈ B.

a = x , b = x + s1y ,

d = x + s2y , c = x + (s1 + s2)y ,

Then g at the points a, b, c , d are

either g(a),−g(a),−g(d), g(d)

or g(a), g(b),−g(b),−g(a),

respectively.Let A and B be such that C \ {0} = C∗ = A ∪∗ B and A = −B.

We define h : C→ {−1, 1} as follows:

h(x) =

{1, if g(x) ∈ A

−1, if g(x) ∈ B.

a = x , b = x + s1y ,

d = x + s2y , c = x + (s1 + s2)y ,

Then g at the points a, b, c , d are

either g(a),−g(a),−g(d), g(d)

or g(a), g(b),−g(b),−g(a),

respectively.Let A and B be such that C \ {0} = C∗ = A ∪∗ B and A = −B.We define h : C→ {−1, 1} as follows:

h(x) =

{1, if g(x) ∈ A

−1, if g(x) ∈ B.

Euclidean Ramsey theory

Color the points of the plane R2 with 2 colors.

Then we have a 2-coloring h of any finitely generated subgroup(G ,+) of the plane such that two-two sides of every parallelogramhave the same color.

Theorem (Shader11)

For every 2-coloring of the plane and every parallelogram H, thereis a congruent copy σ(H) such that at least three of its verticeshas the same color.

For any given parallelogram H, there is a finite witness12 set R.Thus, if the generator set of G contains R, we get acontradiction.

11L. E. Shader, All right triangles are Ramsey in E 2!, Journ. Comb. Theory(A) 20 (1976), 385-389.

12W. H. Gottschalk, Choice functions and Tychonoff’s theorem, Proc. Amer.Math. Soc. 2 (1951), 172.

Euclidean Ramsey theory

Color the points of the plane R2 with 2 colors.Then we have a 2-coloring h of any finitely generated subgroup(G ,+) of the plane such that two-two sides of every parallelogramhave the same color.

Theorem (Shader11)

For every 2-coloring of the plane and every parallelogram H, thereis a congruent copy σ(H) such that at least three of its verticeshas the same color.

For any given parallelogram H, there is a finite witness12 set R.Thus, if the generator set of G contains R, we get acontradiction.

11L. E. Shader, All right triangles are Ramsey in E 2!, Journ. Comb. Theory(A) 20 (1976), 385-389.

12W. H. Gottschalk, Choice functions and Tychonoff’s theorem, Proc. Amer.Math. Soc. 2 (1951), 172.

Euclidean Ramsey theory

Color the points of the plane R2 with 2 colors.Then we have a 2-coloring h of any finitely generated subgroup(G ,+) of the plane such that two-two sides of every parallelogramhave the same color.

Theorem (Shader11)

For every 2-coloring of the plane and every parallelogram H, thereis a congruent copy σ(H) such that at least three of its verticeshas the same color.

For any given parallelogram H, there is a finite witness12 set R.Thus, if the generator set of G contains R, we get acontradiction.

11L. E. Shader, All right triangles are Ramsey in E 2!, Journ. Comb. Theory(A) 20 (1976), 385-389.

12W. H. Gottschalk, Choice functions and Tychonoff’s theorem, Proc. Amer.Math. Soc. 2 (1951), 172.

Euclidean Ramsey theory

Color the points of the plane R2 with 2 colors.Then we have a 2-coloring h of any finitely generated subgroup(G ,+) of the plane such that two-two sides of every parallelogramhave the same color.

Theorem (Shader11)

For every 2-coloring of the plane and every parallelogram H, thereis a congruent copy σ(H) such that at least three of its verticeshas the same color.

For any given parallelogram H, there is a finite witness12 set R.Thus, if the generator set of G contains R, we get acontradiction.

11L. E. Shader, All right triangles are Ramsey in E 2!, Journ. Comb. Theory(A) 20 (1976), 385-389.

12W. H. Gottschalk, Choice functions and Tychonoff’s theorem, Proc. Amer.Math. Soc. 2 (1951), 172.

Easy application on the coloring of the plane

Question (Hadwiger-Nelson’s problem13)

What is the chromatic number χ of the plane?

Theorem

4 ≤ χ(R2) ≤ 7.

We denote the set of forbidden distances by FD. Previously,FD = {1}.Proposition

Let FD1 = {1,√

5+12 }, FD2 = {1,

√2} and FD3 = {1,

√3}.

Then χFDi(R2) > 4 (i = 1, 2, 3).

13https://en.wikipedia.org/wiki/Hadwiger-Nelson problem

Easy application on the coloring of the plane

Question (Hadwiger-Nelson’s problem13)

What is the chromatic number χ of the plane?

Theorem

4 ≤ χ(R2) ≤ 7.

We denote the set of forbidden distances by FD. Previously,FD = {1}.

Proposition

Let FD1 = {1,√

5+12 }, FD2 = {1,

√2} and FD3 = {1,

√3}.

Then χFDi(R2) > 4 (i = 1, 2, 3).

13https://en.wikipedia.org/wiki/Hadwiger-Nelson problem

Easy application on the coloring of the plane

Question (Hadwiger-Nelson’s problem13)

What is the chromatic number χ of the plane?

Theorem

4 ≤ χ(R2) ≤ 7.

We denote the set of forbidden distances by FD. Previously,FD = {1}.Proposition

Let FD1 = {1,√

5+12 }, FD2 = {1,

√2} and FD3 = {1,

√3}.

Then χFDi(R2) > 4 (i = 1, 2, 3).

13https://en.wikipedia.org/wiki/Hadwiger-Nelson problem

Commensurable points in a line

After rescaling the line we can assume that the points are integers:z0 = 0 < z1 < . . . < zk = n ∈ Z.

TheoremThe set D = {z0 = 0, z1, . . . , zk = n} ⊂ C has the weightedPompeiu property.

Proof. Let M be a sufficiently large positive integer,I = {1, . . . , n2 + 1} and consider the following system of triangles:the triangle ∆i ,j ((i , j) ∈ I × I ) is defined such that1. the sides of ∆i ,j are of positive integer length(s) M, ai ,j , bi ,j .2. for any fixed i the number of ai ,1, . . . , ai ,n2+1 are different.3. ai ,j + bi ,j = M + i for every (i , j) ∈ I × I .4. the triangles are lying on the segment [0,M] in the complexplane and the third vertex is contained in the upper half-plane.

Commensurable points in a line

After rescaling the line we can assume that the points are integers:z0 = 0 < z1 < . . . < zk = n ∈ Z.

TheoremThe set D = {z0 = 0, z1, . . . , zk = n} ⊂ C has the weightedPompeiu property.

Proof. Let M be a sufficiently large positive integer,I = {1, . . . , n2 + 1} and consider the following system of triangles:the triangle ∆i ,j ((i , j) ∈ I × I ) is defined such that1. the sides of ∆i ,j are of positive integer length(s) M, ai ,j , bi ,j .2. for any fixed i the number of ai ,1, . . . , ai ,n2+1 are different.3. ai ,j + bi ,j = M + i for every (i , j) ∈ I × I .4. the triangles are lying on the segment [0,M] in the complexplane and the third vertex is contained in the upper half-plane.

Commensurable points in a line

After rescaling the line we can assume that the points are integers:z0 = 0 < z1 < . . . < zk = n ∈ Z.

TheoremThe set D = {z0 = 0, z1, . . . , zk = n} ⊂ C has the weightedPompeiu property.

Proof. Let M be a sufficiently large positive integer,I = {1, . . . , n2 + 1} and consider the following system of triangles:the triangle ∆i ,j ((i , j) ∈ I × I ) is defined such that1. the sides of ∆i ,j are of positive integer length(s) M, ai ,j , bi ,j .2. for any fixed i the number of ai ,1, . . . , ai ,n2+1 are different.3. ai ,j + bi ,j = M + i for every (i , j) ∈ I × I .4. the triangles are lying on the segment [0,M] in the complexplane and the third vertex is contained in the upper half-plane.

Figure: System of triangles; M=12, i=4

RemarkThe length of the major axis is M + i for any fixedi = 1 . . . n2 + 1.

Let us denote the unit vectors parallel to the sides ai ,j and bi ,j in Cby vi ,j and wi ,j , respectively: ai ,j · vi ,j + bi ,j · wi ,j = M · 1.

Anelement y is chosen such that f (y) 6= 0, where f is a non-zerosolution of the Pompeiu equation on the congruent copies of D.Consider the integer grid G∆ generated by the elements

1, vi ,j , wi .j , y .

G∆ is a group such that all of the elements are linear combinationswith integer coefficients of the generating vectors.

Let us denote the unit vectors parallel to the sides ai ,j and bi ,j in Cby vi ,j and wi ,j , respectively: ai ,j · vi ,j + bi ,j · wi ,j = M · 1. Anelement y is chosen such that f (y) 6= 0, where f is a non-zerosolution of the Pompeiu equation on the congruent copies of D.

Consider the integer grid G∆ generated by the elements

1, vi ,j , wi .j , y .

G∆ is a group such that all of the elements are linear combinationswith integer coefficients of the generating vectors.

Let us denote the unit vectors parallel to the sides ai ,j and bi ,j in Cby vi ,j and wi ,j , respectively: ai ,j · vi ,j + bi ,j · wi ,j = M · 1. Anelement y is chosen such that f (y) 6= 0, where f is a non-zerosolution of the Pompeiu equation on the congruent copies of D.Consider the integer grid G∆ generated by the elements

1, vi ,j , wi .j , y .

G∆ is a group such that all of the elements are linear combinationswith integer coefficients of the generating vectors.

Since G∆ is finitely generated, spectral analysis holds in thevarieties of the vector space of complex valued functions definedon G∆.

This means that the variety formed by the solutions of thePompeiu equation

k∑i=0

ai f (σ(zi )) = 0 (σ ∈ Iso(G∆)) (8)

on G∆ contains a non-zero exponential element g : G∆ → C, i.e.

k∑i=0

aig(σ(zi )) = 0 (σ ∈ Iso(G∆)).

If σ = 1 then the exponential property shows that

k∑i=0

ai (g(1))zi = 0. (9)

i.e. g(1) is the root of the polynomialP(x) = a0 · 1 + a1 · xz1 + . . .+ ak · xzk .Similarly, the same is true for g(vi ,j) and g(wi ,j).

Since G∆ is finitely generated, spectral analysis holds in thevarieties of the vector space of complex valued functions definedon G∆. This means that the variety formed by the solutions of thePompeiu equation

k∑i=0

ai f (σ(zi )) = 0 (σ ∈ Iso(G∆)) (8)

on G∆ contains a non-zero exponential element g : G∆ → C, i.e.

k∑i=0

aig(σ(zi )) = 0 (σ ∈ Iso(G∆)).

If σ = 1 then the exponential property shows that

k∑i=0

ai (g(1))zi = 0. (9)

i.e. g(1) is the root of the polynomialP(x) = a0 · 1 + a1 · xz1 + . . .+ ak · xzk .Similarly, the same is true for g(vi ,j) and g(wi ,j).

Since G∆ is finitely generated, spectral analysis holds in thevarieties of the vector space of complex valued functions definedon G∆. This means that the variety formed by the solutions of thePompeiu equation

k∑i=0

ai f (σ(zi )) = 0 (σ ∈ Iso(G∆)) (8)

on G∆ contains a non-zero exponential element g : G∆ → C, i.e.

k∑i=0

aig(σ(zi )) = 0 (σ ∈ Iso(G∆)).

If σ = 1 then the exponential property shows that

k∑i=0

ai (g(1))zi = 0. (9)

i.e. g(1) is the root of the polynomialP(x) = a0 · 1 + a1 · xz1 + . . .+ ak · xzk .Similarly, the same is true for g(vi ,j) and g(wi ,j).

Since G∆ is finitely generated, spectral analysis holds in thevarieties of the vector space of complex valued functions definedon G∆. This means that the variety formed by the solutions of thePompeiu equation

k∑i=0

ai f (σ(zi )) = 0 (σ ∈ Iso(G∆)) (8)

on G∆ contains a non-zero exponential element g : G∆ → C, i.e.

k∑i=0

aig(σ(zi )) = 0 (σ ∈ Iso(G∆)).

If σ = 1 then the exponential property shows that

k∑i=0

ai (g(1))zi = 0. (9)

i.e. g(1) is the root of the polynomialP(x) = a0 · 1 + a1 · xz1 + . . .+ ak · xzk .

Similarly, the same is true for g(vi ,j) and g(wi ,j).

Since G∆ is finitely generated, spectral analysis holds in thevarieties of the vector space of complex valued functions definedon G∆. This means that the variety formed by the solutions of thePompeiu equation

k∑i=0

ai f (σ(zi )) = 0 (σ ∈ Iso(G∆)) (8)

on G∆ contains a non-zero exponential element g : G∆ → C, i.e.

k∑i=0

aig(σ(zi )) = 0 (σ ∈ Iso(G∆)).

If σ = 1 then the exponential property shows that

k∑i=0

ai (g(1))zi = 0. (9)

i.e. g(1) is the root of the polynomialP(x) = a0 · 1 + a1 · xz1 + . . .+ ak · xzk .Similarly, the same is true for g(vi ,j) and g(wi ,j).

LemmaThe root g(1) must be a root of unity.

Proof. Consider the pairs (g(vi ,j), g(wi ,j)).

Since the maximal number of the pairs is n2 buti , j ∈ {1, . . . , n2 + 1}, using the pigeon-hole principle we get one ofthe pairs of the roots is taken at least two times in each ”row”.Repeating the argument for the ”column” we have that thereexists a pair of roots belonging to at least four triangles. For thesake of simplicity let λ1, . . . , λn be the roots of

P(x) = a0 · 1 + a1 · xz1 + . . .+ ak · xzk ,

g(1) = λ1 and suppose that (λ2, λ3) is the pair of the roots alongthe (non-horizontal) directions chosen above.

LemmaThe root g(1) must be a root of unity.

Proof. Consider the pairs (g(vi ,j), g(wi ,j)).Since the maximal number of the pairs is n2 buti , j ∈ {1, . . . , n2 + 1}, using the pigeon-hole principle we get one ofthe pairs of the roots is taken at least two times in each ”row”.Repeating the argument for the ”column” we have that thereexists a pair of roots belonging to at least four triangles.

For thesake of simplicity let λ1, . . . , λn be the roots of

P(x) = a0 · 1 + a1 · xz1 + . . .+ ak · xzk ,

g(1) = λ1 and suppose that (λ2, λ3) is the pair of the roots alongthe (non-horizontal) directions chosen above.

LemmaThe root g(1) must be a root of unity.

Proof. Consider the pairs (g(vi ,j), g(wi ,j)).Since the maximal number of the pairs is n2 buti , j ∈ {1, . . . , n2 + 1}, using the pigeon-hole principle we get one ofthe pairs of the roots is taken at least two times in each ”row”.Repeating the argument for the ”column” we have that thereexists a pair of roots belonging to at least four triangles. For thesake of simplicity let λ1, . . . , λn be the roots of

P(x) = a0 · 1 + a1 · xz1 + . . .+ ak · xzk ,

g(1) = λ1 and suppose that (λ2, λ3) is the pair of the roots alongthe (non-horizontal) directions chosen above.

This means that

g(vi ,j1) = g(vi ,j2) = g(vi ′,j ′1) = g(vi ′,j ′2) = λ2,

g(wi ,j1) = g(wi ,j2) = g(wi ′,j ′1) = g(wi ′,j ′2

) = λ3.

Since g is exponential and ai ,jvi ,j + bi ,jwi ,j = M we have

λM1 = λai,j12 λ

bi,j13 = λ

ai,j22 λ

bi,j23 ,

λM1 = λai′,j′

12 λ

bi′,j′1

3 = λai′,j′

22 λ

bi′,j′2

3 .

Using that ai ,j1 + bi ,j1 = ai ,j2 + bi ,j2 = M + i , these equations implythat λ1 is a root of unity.

Corollary

The polynomial P(x) = a0 · 1 + a1 · xz1 + . . .+ ak · xzk has a rootof unity.

There is an N ∈ Z+ such that (λk)N = 1 for each root λk of P(x)which is a root of unity.

This means that

g(vi ,j1) = g(vi ,j2) = g(vi ′,j ′1) = g(vi ′,j ′2) = λ2,

g(wi ,j1) = g(wi ,j2) = g(wi ′,j ′1) = g(wi ′,j ′2

) = λ3.

Since g is exponential and ai ,jvi ,j + bi ,jwi ,j = M we have

λM1 = λai,j12 λ

bi,j13 = λ

ai,j22 λ

bi,j23 ,

λM1 = λai′,j′

12 λ

bi′,j′1

3 = λai′,j′

22 λ

bi′,j′2

3 .

Using that ai ,j1 + bi ,j1 = ai ,j2 + bi ,j2 = M + i , these equations implythat λ1 is a root of unity.

Corollary

The polynomial P(x) = a0 · 1 + a1 · xz1 + . . .+ ak · xzk has a rootof unity.

There is an N ∈ Z+ such that (λk)N = 1 for each root λk of P(x)which is a root of unity.

This means that

g(vi ,j1) = g(vi ,j2) = g(vi ′,j ′1) = g(vi ′,j ′2) = λ2,

g(wi ,j1) = g(wi ,j2) = g(wi ′,j ′1) = g(wi ′,j ′2

) = λ3.

Since g is exponential and ai ,jvi ,j + bi ,jwi ,j = M we have

λM1 = λai,j12 λ

bi,j13 = λ

ai,j22 λ

bi,j23 ,

λM1 = λai′,j′

12 λ

bi′,j′1

3 = λai′,j′

22 λ

bi′,j′2

3 .

Using that ai ,j1 + bi ,j1 = ai ,j2 + bi ,j2 = M + i , these equations implythat λ1 is a root of unity.

Corollary

The polynomial P(x) = a0 · 1 + a1 · xz1 + . . .+ ak · xzk has a rootof unity.

There is an N ∈ Z+ such that (λk)N = 1 for each root λk of P(x)which is a root of unity.

This means that

g(vi ,j1) = g(vi ,j2) = g(vi ′,j ′1) = g(vi ′,j ′2) = λ2,

g(wi ,j1) = g(wi ,j2) = g(wi ′,j ′1) = g(wi ′,j ′2

) = λ3.

Since g is exponential and ai ,jvi ,j + bi ,jwi ,j = M we have

λM1 = λai,j12 λ

bi,j13 = λ

ai,j22 λ

bi,j23 ,

λM1 = λai′,j′

12 λ

bi′,j′1

3 = λai′,j′

22 λ

bi′,j′2

3 .

Using that ai ,j1 + bi ,j1 = ai ,j2 + bi ,j2 = M + i , these equations implythat λ1 is a root of unity.

Corollary

The polynomial P(x) = a0 · 1 + a1 · xz1 + . . .+ ak · xzk has a rootof unity.

There is an N ∈ Z+ such that (λk)N = 1 for each root λk of P(x)which is a root of unity.

End of the proofConsider the original sample z0 = 0, z1, . . . , zk = n. Choose Msuch that N|M, i.e. λMk = 1.

Draw an isosceles triangles Ti with side lengths M,M, 1 indicatedby the vectors v1, v2, 1.

Extending G∆, similarly as above we get that g(vi ) is a root of thepolynomial

P(x) = a0 · 1 + a1 · xz1 + . . .+ ak · xzk

and also a root of unity. By the choice of M and since g isexponential

g(M · vj) = [g(vj)]M = 1 (j = 1, 2). (10)

This implies that 1 = g(0) = g(z0) = · · · = g(zk) whichcontradicts to

∑ki=0 aig(zi ) = 0, where

∑ki=0 ai 6= 0.

Extending G∆, similarly as above we get that g(vi ) is a root of thepolynomial

P(x) = a0 · 1 + a1 · xz1 + . . .+ ak · xzk

and also a root of unity. By the choice of M and since g isexponential

g(M · vj) = [g(vj)]M = 1 (j = 1, 2). (10)

This implies that 1 = g(0) = g(z0) = · · · = g(zk) whichcontradicts to

∑ki=0 aig(zi ) = 0, where

∑ki=0 ai 6= 0.

Extending G∆, similarly as above we get that g(vi ) is a root of thepolynomial

P(x) = a0 · 1 + a1 · xz1 + . . .+ ak · xzk

and also a root of unity. By the choice of M and since g isexponential

g(M · vj) = [g(vj)]M = 1 (j = 1, 2). (10)

This implies that 1 = g(0) = g(z0) = · · · = g(zk) whichcontradicts to

∑ki=0 aig(zi ) = 0, where

∑ki=0 ai 6= 0.

A one-parameter family of quadrangles with thediscrete Pompeiu property

Let ABC4 be a non-isosceles triangle and C and B be on thesame side of the perpendicular bisector of AB. Let 0 < α < 45◦ bea given angle and choose a point P on the line AB as in the figurebelow.

LemmaFor any given angle 0 < α < 45◦ the set H = {A,B,C ,D} has thePompeiu property.

Proof. For any point X let X ′ be the image of X under thereflection on the perpendicular bisector of AB. Then A′ = B,B ′ = A and the points C , C ′, D and D ′ form a symmetrictrapezium such that D ′C = CC ′ = C ′D.

Taking the difference of equations

f (A)+f (B)+f (C )+f (D) = 0 and f (A′)+f (B ′)+f (C ′)+f (D ′) = 0.

It follows that

f (C )− f (C ′) + f (D)− f (D ′) = 0. (11)

Since equation (11) holds on any congruent copy of the trapeziumCC ′DD ′ we have

f (C )−f (C ′)+f (D)−f (D ′) = 0 and f (C )−f (C ′)+f (D)−f (H ′) = 0.(12)

This implies that f (D ′) = f (H ′).

Taking the difference of equations

f (A)+f (B)+f (C )+f (D) = 0 and f (A′)+f (B ′)+f (C ′)+f (D ′) = 0.

It follows that

f (C )− f (C ′) + f (D)− f (D ′) = 0. (11)

Since equation (11) holds on any congruent copy of the trapeziumCC ′DD ′ we have

f (C )−f (C ′)+f (D)−f (D ′) = 0 and f (C )−f (C ′)+f (D)−f (H ′) = 0.(12)

This implies that f (D ′) = f (H ′).

Taking the difference of equations

f (A)+f (B)+f (C )+f (D) = 0 and f (A′)+f (B ′)+f (C ′)+f (D ′) = 0.

It follows that

f (C )− f (C ′) + f (D)− f (D ′) = 0. (11)

Since equation (11) holds on any congruent copy of the trapeziumCC ′DD ′ we have

f (C )−f (C ′)+f (D)−f (D ′) = 0 and f (C )−f (C ′)+f (D)−f (H ′) = 0.(12)

This implies that f (D ′) = f (H ′).

Group acting on R2 Set P WP

Translations finite sets with at least 2 elements No No

Rigid motionssets with at most 3 elements Yes ?parallelograms Yes ?finite sets of collinear points Yes Yeswith commensurable distances

Isometriesnon-collinear sets Yes Yeswith at most 3 elementsrational 4-point sets Yes YesPompeiu quadrangles Yes ?

Direct similarities all finite sets Yes Yes

Thank you for your kind attention.